Step | Hyp | Ref
| Expression |
1 | | ssun2 3739 |
. . . 4
⊢ {(𝑋𝐹(𝑊‘𝑋))} ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
2 | | fpwwe2.1 |
. . . . . . . . . . . . . 14
⊢ 𝑊 = {〈𝑥, 𝑟〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} |
3 | | fpwwe2.2 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ∈ V) |
4 | 3 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝐴 ∈ V) |
5 | | fpwwe2.3 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
6 | 5 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
7 | | fpwwe2.4 |
. . . . . . . . . . . . . 14
⊢ 𝑋 = ∪
dom 𝑊 |
8 | 2, 4, 6, 7 | fpwwe2lem12 9342 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋 ∈ dom 𝑊) |
9 | 2, 4, 6, 7 | fpwwe2lem11 9341 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋)) |
10 | | ffun 5961 |
. . . . . . . . . . . . . 14
⊢ (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊) |
11 | | funfvbrb 6238 |
. . . . . . . . . . . . . 14
⊢ (Fun
𝑊 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
12 | 9, 10, 11 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
13 | 8, 12 | mpbid 221 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋𝑊(𝑊‘𝑋)) |
14 | 2, 4 | fpwwe2lem2 9333 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝑊(𝑊‘𝑋) ↔ ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
15 | 13, 14 | mpbid 221 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))) |
16 | 15 | simpld 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋))) |
17 | 16 | simpld 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋 ⊆ 𝐴) |
18 | 16 | simprd 478 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
19 | 15 | simprd 478 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)) |
20 | 19 | simpld 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) We 𝑋) |
21 | 17, 18, 20 | 3jca 1235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ∧ (𝑊‘𝑋) We 𝑋)) |
22 | 2, 3, 5 | fpwwe2lem5 9335 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ∧ (𝑊‘𝑋) We 𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝐴) |
23 | 21, 22 | syldan 486 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝐴) |
24 | 23 | snssd 4281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝐴) |
25 | 17, 24 | unssd 3751 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴) |
26 | | ssun1 3738 |
. . . . . . . . . . 11
⊢ 𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
27 | | xpss12 5148 |
. . . . . . . . . . 11
⊢ ((𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 × 𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
28 | 26, 26, 27 | mp2an 704 |
. . . . . . . . . 10
⊢ (𝑋 × 𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
29 | 18, 28 | syl6ss 3580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
30 | | xpss12 5148 |
. . . . . . . . . . 11
⊢ ((𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ {(𝑋𝐹(𝑊‘𝑋))} ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
31 | 26, 1, 30 | mp2an 704 |
. . . . . . . . . 10
⊢ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
32 | 31 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
33 | 29, 32 | unssd 3751 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
34 | 25, 33 | jca 553 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})))) |
35 | | ssdif0 3896 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} ↔ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
36 | | simpllr 795 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
37 | 18 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
38 | 37 | ssbrd 4626 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)))) |
39 | | brxp 5071 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
40 | 39 | simplbi 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
41 | 38, 40 | syl6 34 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
42 | 36, 41 | mtod 188 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋))) |
43 | | brxp 5071 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
44 | 43 | simplbi 475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
45 | 36, 44 | nsyl 134 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))) |
46 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑋𝐹(𝑊‘𝑋)) ∈ V |
47 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))(𝑋𝐹(𝑊‘𝑋)))) |
48 | | brun 4633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
49 | 47, 48 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))))) |
50 | 49 | notbid 307 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → (¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))))) |
51 | 46, 50 | rexsn 4170 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑦 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
52 | | ioran 510 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))) ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
53 | 51, 52 | bitri 263 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑦 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
54 | 42, 45, 53 | sylanbrc 695 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ∃𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
55 | | simplrr 797 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 ≠ ∅) |
56 | 55 | neneqd 2787 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑥 = ∅) |
57 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) |
58 | | sssn 4298 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} ↔ (𝑥 = ∅ ∨ 𝑥 = {(𝑋𝐹(𝑊‘𝑋))})) |
59 | 57, 58 | sylib 207 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥 = ∅ ∨ 𝑥 = {(𝑋𝐹(𝑊‘𝑋))})) |
60 | 59 | ord 391 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (¬ 𝑥 = ∅ → 𝑥 = {(𝑋𝐹(𝑊‘𝑋))})) |
61 | 56, 60 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 = {(𝑋𝐹(𝑊‘𝑋))}) |
62 | 61 | raleqdv 3121 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
63 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
64 | 63 | notbid 307 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
65 | 46, 64 | ralsn 4169 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
66 | 62, 65 | syl6bb 275 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
67 | 61, 66 | rexeqbidv 3130 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ∃𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
68 | 54, 67 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
69 | 68 | ex 449 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
70 | 35, 69 | syl5bir 232 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
71 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑥 ∈ V |
72 | | difexg 4735 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ V → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
73 | 71, 72 | mp1i 13 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
74 | | wefr 5028 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Fr 𝑋) |
75 | 20, 74 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) Fr 𝑋) |
76 | 75 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑊‘𝑋) Fr 𝑋) |
77 | | simplrl 796 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → 𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
78 | | uncom 3719 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋) |
79 | 77, 78 | syl6sseq 3614 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → 𝑥 ⊆ ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋)) |
80 | | ssundif 4004 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ⊆ ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋) ↔ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
81 | 79, 80 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
82 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) |
83 | | fri 5000 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋 ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅)) → ∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) |
84 | 73, 76, 81, 82, 83 | syl22anc 1319 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) |
85 | | brun 4633 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
86 | | idd 24 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧(𝑊‘𝑋)𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
87 | | brxp 5071 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
88 | 87 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
89 | | eldifn 3695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
90 | 89 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
91 | 90 | pm2.21d 117 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑧(𝑊‘𝑋)𝑦)) |
92 | 88, 91 | syl5 33 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
93 | 86, 92 | jaod 394 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) → 𝑧(𝑊‘𝑋)𝑦)) |
94 | 85, 93 | syl5bi 231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
95 | 94 | con3d 147 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (¬ 𝑧(𝑊‘𝑋)𝑦 → ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
96 | 95 | ralimdv 2946 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
97 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
98 | 97 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
99 | 18 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
100 | 99 | ssbrd 4626 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦)) |
101 | | brxp 5071 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) |
102 | 101 | simplbi 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
103 | 100, 102 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
104 | 98, 103 | mtod 188 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦) |
105 | | brxp 5071 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
106 | 105 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
107 | 90, 106 | nsyl 134 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
108 | | brun 4633 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
109 | 63, 108 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦))) |
110 | 109 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦))) |
111 | 46, 110 | ralsn 4169 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
112 | | ioran 510 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
113 | 111, 112 | bitri 263 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
114 | 104, 107,
113 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
115 | 96, 114 | jctird 565 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
116 | | ssun1 3738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑥 ⊆ (𝑥 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
117 | | undif1 3995 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) = (𝑥 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
118 | 116, 117 | sseqtr4i 3601 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑥 ⊆ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
119 | | ralun 3757 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑧 ∈
(𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) → ∀𝑧 ∈ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
120 | | ssralv 3629 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ⊆ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
121 | 118, 119,
120 | mpsyl 66 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑧 ∈
(𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
122 | 115, 121 | syl6 34 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
123 | | eldifi 3694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑦 ∈ 𝑥) |
124 | 123 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → 𝑦 ∈ 𝑥) |
125 | 122, 124 | jctild 564 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
126 | 125 | expimpd 627 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ((𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) → (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
127 | 126 | reximdv2 2997 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
128 | 84, 127 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
129 | 128 | ex 449 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
130 | 70, 129 | pm2.61dne 2868 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
131 | 130 | ex 449 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
132 | 131 | alrimiv 1842 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑥((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
133 | | df-fr 4997 |
. . . . . . . . . 10
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ ∀𝑥((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
134 | 132, 133 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
135 | | elun 3715 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
136 | | elun 3715 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
137 | 135, 136 | anbi12i 729 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) ↔ ((𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}))) |
138 | | weso 5029 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Or 𝑋) |
139 | 20, 138 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) Or 𝑋) |
140 | | solin 4982 |
. . . . . . . . . . . . . . 15
⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥)) |
141 | 139, 140 | sylan 487 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥)) |
142 | | ssun1 3738 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊‘𝑋) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
143 | 142 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑊‘𝑋) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))) |
144 | 143 | ssbrd 4626 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 → 𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
145 | | idd 24 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = 𝑦 → 𝑥 = 𝑦)) |
146 | 143 | ssbrd 4626 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦(𝑊‘𝑋)𝑥 → 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
147 | 144, 145,
146 | 3orim123d 1399 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
148 | 141, 147 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
149 | 148 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
150 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) |
151 | 150 | ancomd 466 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
152 | | brxp 5071 |
. . . . . . . . . . . . . . 15
⊢ (𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
153 | 151, 152 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → 𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥) |
154 | | ssun2 3739 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
155 | 154 | ssbri 4627 |
. . . . . . . . . . . . . 14
⊢ (𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥 → 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥) |
156 | | 3mix3 1225 |
. . . . . . . . . . . . . 14
⊢ (𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥 → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
157 | 153, 155,
156 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
158 | 157 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
159 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
160 | | brxp 5071 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
161 | 159, 160 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → 𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
162 | 154 | ssbri 4627 |
. . . . . . . . . . . . . 14
⊢ (𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
163 | | 3mix1 1223 |
. . . . . . . . . . . . . 14
⊢ (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
164 | 161, 162,
163 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
165 | 164 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
166 | | elsni 4142 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑥 = (𝑋𝐹(𝑊‘𝑋))) |
167 | | elsni 4142 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
168 | | eqtr3 2631 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = (𝑋𝐹(𝑊‘𝑋)) ∧ 𝑦 = (𝑋𝐹(𝑊‘𝑋))) → 𝑥 = 𝑦) |
169 | 166, 167,
168 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 = 𝑦) |
170 | 169 | 3mix2d 1230 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
171 | 170 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
172 | 149, 158,
165, 171 | ccased 985 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (((𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
173 | 137, 172 | syl5bi 231 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
174 | 173 | ralrimivv 2953 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})(𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
175 | | dfwe2 6873 |
. . . . . . . . 9
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})(𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
176 | 134, 174,
175 | sylanbrc 695 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
177 | 2 | fpwwe2cbv 9331 |
. . . . . . . . . . . . 13
⊢ 𝑊 = {〈𝑎, 𝑠〉 ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑏](𝑏𝐹(𝑠 ∩ (𝑏 × 𝑏))) = 𝑧))} |
178 | 177, 4, 13 | fpwwe2lem3 9334 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦) |
179 | | cnvimass 5404 |
. . . . . . . . . . . . . . 15
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ⊆ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
180 | | fvex 6113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊‘𝑋) ∈ V |
181 | | snex 4835 |
. . . . . . . . . . . . . . . . . 18
⊢ {(𝑋𝐹(𝑊‘𝑋))} ∈ V |
182 | | xpexg 6858 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋 ∈ dom 𝑊 ∧ {(𝑋𝐹(𝑊‘𝑋))} ∈ V) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
183 | 8, 181, 182 | sylancl 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
184 | | unexg 6857 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑊‘𝑋) ∈ V ∧ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
185 | 180, 183,
184 | sylancr 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
186 | | dmexg 6989 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V → dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
187 | 185, 186 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
188 | | ssexg 4732 |
. . . . . . . . . . . . . . 15
⊢ (((◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ⊆ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∧ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
189 | 179, 187,
188 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
190 | 189 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
191 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) → 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) |
192 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) |
193 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
194 | | nelne2 2879 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ 𝑋 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑦 ≠ (𝑋𝐹(𝑊‘𝑋))) |
195 | 192, 193,
194 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ≠ (𝑋𝐹(𝑊‘𝑋))) |
196 | 88, 167 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
197 | 196 | necon3ai 2807 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ≠ (𝑋𝐹(𝑊‘𝑋)) → ¬ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
198 | | biorf 419 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → (𝑧(𝑊‘𝑋)𝑦 ↔ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦))) |
199 | 195, 197,
198 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧(𝑊‘𝑋)𝑦 ↔ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦))) |
200 | | orcom 401 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦) ↔ (𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
201 | 200, 85 | bitr4i 266 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
202 | 199, 201 | syl6rbb 276 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ 𝑧(𝑊‘𝑋)𝑦)) |
203 | | vex 3176 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑦 ∈ V |
204 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑧 ∈ V |
205 | 204 | eliniseg 5413 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ V → (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
206 | 203, 205 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
207 | 204 | eliniseg 5413 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ V → (𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}) ↔ 𝑧(𝑊‘𝑋)𝑦)) |
208 | 203, 207 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}) ↔ 𝑧(𝑊‘𝑋)𝑦) |
209 | 202, 206,
208 | 3bitr4g 302 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}))) |
210 | 209 | eqrdv 2608 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = (◡(𝑊‘𝑋) “ {𝑦})) |
211 | 191, 210 | sylan9eqr 2666 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → 𝑢 = (◡(𝑊‘𝑋) “ {𝑦})) |
212 | 211 | sqxpeqd 5065 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢 × 𝑢) = ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) |
213 | 212 | ineq2d 3776 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
214 | | indir 3834 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
215 | | inxp 5176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) |
216 | | incom 3767 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦})) = ((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) |
217 | | cnvimass 5404 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (◡(𝑊‘𝑋) “ {𝑦}) ⊆ dom (𝑊‘𝑋) |
218 | 18 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
219 | | dmss 5245 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
220 | 218, 219 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
221 | | dmxpid 5266 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
222 | 220, 221 | syl6sseq 3614 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → dom (𝑊‘𝑋) ⊆ 𝑋) |
223 | 217, 222 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡(𝑊‘𝑋) “ {𝑦}) ⊆ 𝑋) |
224 | 223, 193 | ssneldd 3571 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ (◡(𝑊‘𝑋) “ {𝑦})) |
225 | | disjsn 4192 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ ↔ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ (◡(𝑊‘𝑋) “ {𝑦})) |
226 | 224, 225 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
227 | 216, 226 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦})) = ∅) |
228 | 227 | xpeq2d 5063 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ∅)) |
229 | | xp0 5471 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ∅) =
∅ |
230 | 228, 229 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) = ∅) |
231 | 215, 230 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ∅) |
232 | 231 | uneq2d 3729 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅)) |
233 | 214, 232 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅)) |
234 | | un0 3919 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) |
235 | 233, 234 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
236 | 235 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
237 | 213, 236 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
238 | 211, 237 | oveq12d 6567 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))))) |
239 | 238 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → ((𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦)) |
240 | 190, 239 | sbcied 3439 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ([(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦)) |
241 | 178, 240 | mpbird 246 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
242 | 167 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
243 | 242 | eqcomd 2616 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋𝐹(𝑊‘𝑋)) = 𝑦) |
244 | 189 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
245 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
246 | 242 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑦 ∈ dom ◡(𝑊‘𝑋) ↔ (𝑋𝐹(𝑊‘𝑋)) ∈ dom ◡(𝑊‘𝑋))) |
247 | 18 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
248 | | rnss 5275 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → ran (𝑊‘𝑋) ⊆ ran (𝑋 × 𝑋)) |
249 | 247, 248 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ran (𝑊‘𝑋) ⊆ ran (𝑋 × 𝑋)) |
250 | | df-rn 5049 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
(𝑊‘𝑋) = dom ◡(𝑊‘𝑋) |
251 | | rnxpid 5486 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
(𝑋 × 𝑋) = 𝑋 |
252 | 249, 250,
251 | 3sstr3g 3608 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → dom ◡(𝑊‘𝑋) ⊆ 𝑋) |
253 | 252 | sseld 3567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋)) ∈ dom ◡(𝑊‘𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
254 | 246, 253 | sylbid 229 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑦 ∈ dom ◡(𝑊‘𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
255 | 245, 254 | mtod 188 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑦 ∈ dom ◡(𝑊‘𝑋)) |
256 | | ndmima 5421 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬
𝑦 ∈ dom ◡(𝑊‘𝑋) → (◡(𝑊‘𝑋) “ {𝑦}) = ∅) |
257 | 255, 256 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑊‘𝑋) “ {𝑦}) = ∅) |
258 | 242 | sneqd 4137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → {𝑦} = {(𝑋𝐹(𝑊‘𝑋))}) |
259 | 258 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦}) = (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))})) |
260 | | df-ima 5051 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))}) = ran (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) |
261 | | cnvxp 5470 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
262 | 261 | reseq1i 5313 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) |
263 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {(𝑋𝐹(𝑊‘𝑋))} ⊆ {(𝑋𝐹(𝑊‘𝑋))} |
264 | | xpssres 5354 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ⊆ {(𝑋𝐹(𝑊‘𝑋))} → (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋)) |
265 | 263, 264 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
266 | 262, 265 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
267 | 266 | rneqi 5273 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
(◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ran ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
268 | 46 | snnz 4252 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {(𝑋𝐹(𝑊‘𝑋))} ≠ ∅ |
269 | | rnxp 5483 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ≠ ∅ → ran ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) = 𝑋) |
270 | 268, 269 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) = 𝑋 |
271 | 267, 270 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ran
(◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = 𝑋 |
272 | 260, 271 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))}) = 𝑋 |
273 | 259, 272 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦}) = 𝑋) |
274 | 257, 273 | uneq12d 3730 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) = (∅ ∪ 𝑋)) |
275 | | cnvun 5457 |
. . . . . . . . . . . . . . . . . . 19
⊢ ◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) = (◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
276 | 275 | imaeq1i 5382 |
. . . . . . . . . . . . . . . . . 18
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) |
277 | | imaundir 5465 |
. . . . . . . . . . . . . . . . . 18
⊢ ((◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) |
278 | 276, 277 | eqtri 2632 |
. . . . . . . . . . . . . . . . 17
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) |
279 | | un0 3919 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ ∅) = 𝑋 |
280 | | uncom 3719 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ ∅) = (∅ ∪
𝑋) |
281 | 279, 280 | eqtr3i 2634 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑋 = (∅ ∪ 𝑋) |
282 | 274, 278,
281 | 3eqtr4g 2669 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = 𝑋) |
283 | 191, 282 | sylan9eqr 2666 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → 𝑢 = 𝑋) |
284 | 283 | sqxpeqd 5065 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢 × 𝑢) = (𝑋 × 𝑋)) |
285 | 284 | ineq2d 3776 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋))) |
286 | | indir 3834 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋))) |
287 | | df-ss 3554 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ↔ ((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
288 | 247, 287 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
289 | | incom 3767 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋) = (𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) |
290 | | disjsn 4192 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ ↔ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
291 | 245, 290 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
292 | 289, 291 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋) = ∅) |
293 | 292 | xpeq2d 5063 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋 × ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋)) = (𝑋 × ∅)) |
294 | | xpindi 5177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 × ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋)) = ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋)) |
295 | | xp0 5471 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 × ∅) =
∅ |
296 | 293, 294,
295 | 3eqtr3g 2667 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋)) = ∅) |
297 | 288, 296 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋))) = ((𝑊‘𝑋) ∪ ∅)) |
298 | 286, 297 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = ((𝑊‘𝑋) ∪ ∅)) |
299 | | un0 3919 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊‘𝑋) ∪ ∅) = (𝑊‘𝑋) |
300 | 298, 299 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
301 | 300 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
302 | 285, 301 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (𝑊‘𝑋)) |
303 | 283, 302 | oveq12d 6567 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = (𝑋𝐹(𝑊‘𝑋))) |
304 | 303 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → ((𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑋𝐹(𝑊‘𝑋)) = 𝑦)) |
305 | 244, 304 | sbcied 3439 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ([(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑋𝐹(𝑊‘𝑋)) = 𝑦)) |
306 | 243, 305 | mpbird 246 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
307 | 241, 306 | jaodan 822 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
308 | 136, 307 | sylan2b 491 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
309 | 308 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
310 | 176, 309 | jca 553 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)) |
311 | 2, 3 | fpwwe2lem2 9333 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ↔ (((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) ∧ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
312 | 311 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ↔ (((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) ∧ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
313 | 34, 310, 312 | mpbir2and 959 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))) |
314 | 2 | relopabi 5167 |
. . . . . . 7
⊢ Rel 𝑊 |
315 | 314 | releldmi 5283 |
. . . . . 6
⊢ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∈ dom 𝑊) |
316 | | elssuni 4403 |
. . . . . 6
⊢ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∈ dom 𝑊 → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ∪
dom 𝑊) |
317 | 313, 315,
316 | 3syl 18 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ∪
dom 𝑊) |
318 | 317, 7 | syl6sseqr 3615 |
. . . 4
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
319 | 1, 318 | syl5ss 3579 |
. . 3
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝑋) |
320 | 46 | snss 4259 |
. . 3
⊢ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ↔ {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝑋) |
321 | 319, 320 | sylibr 223 |
. 2
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
322 | 321 | pm2.18da 458 |
1
⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |